Upper bounds for GCD sums of the form ${\sum }_{k,\ell =1}^N\frac{{\left(\gcd (n_k,n_{\ell })\right)}^{2\alpha }}{{\left(n_k{n}_{\ell }\right)}^{\alpha }}$ are established, where ${\left(n_k\right)}_{1\le k\le N}$ is any sequence of distinct positive integers and $0<\alpha \le 1$; the estimate for $\alpha =1/2$ solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for $\alpha =1/2$. The method of proof is based on identifying the sum as a certain Poisson integral on a polydisc; as a byproduct, estimates for the largest eigenvalues of the associated GCD matrices are also found. The bounds for such GCD sums are used to establish...

By a general Franklin system corresponding to a dense sequence of knots 𝓣 = (tₙ, n ≥ 0) in [0,1] we mean a sequence of orthonormal piecewise linear functions with knots 𝓣, that is, the nth function of the system has knots t₀, ..., tₙ. The main result of this paper is a characterization of sequences 𝓣 for which the corresponding general Franklin system is a basis or an unconditional basis in H¹[0,1].

Cet article est consacré à l’étude des espaces $A_{\omega }=ℱL^2({\mathbf{R}}^n;\omega \left(x\right)dx)$ qui sont des algèbres de Banach. On démontre que les multiplicateurs ponctuels de $A_{\omega }$ sont les fonctions qui appartiennent localement et uniformément à $A_{\omega }$ si et seulement si $A_{\omega }$ contient des fonctions à support compact.

Some extensions of the properties of invariant polynomials proved by Davis (1980), Chikuse (1980), Chikuse and Davis (1986) and Ratnarajah et al. (2005) are given for symmetric and Hermitian matrices.

We consider the Vilenkin orthonormal system on a Vilenkin group $G$ and the Vilenkin-Fourier coefficients $\widehat{f}\left(n\right)$, $n\in \mathbb{N}$, of functions $f\in L^p\left(G\right)$ for some $1<p\le 2$. We obtain certain sufficient conditions for the finiteness of the series ${\sum }_{n=1}^{\infty }{a}_n{\left|\widehat{f}\left(n\right)\right|}^r$, where $\left\{a_n\right\}$ is a given sequence of positive real numbers satisfying a mild assumption and $0<r<2$. We also find analogous conditions for the double Vilenkin-Fourier series. These sufficient conditions are in terms of (either global or local) moduli of continuity of $f$ and give multiplicative analogue...

We introduce the notion of a $g$-atomic subspace for a bounded linear operator and construct several useful resolutions of the identity operator on a Hilbert space using the theory of $g$-fusion frames. Also, we shall describe the concept of frame operator for a pair of $g$-fusion Bessel sequences and some of their properties.

In this paper we study generalized Besov type spaces on the Laguerre hypergroup and we give some characterizations using different equivalent norms which allows to reach results of completeness, continuous embeddings and density of some subspaces. A generalized Calderón-Zygmund formula adapted to the harmonic analysis on the Laguerre Hypergroup is obtained inducing two more equivalent norms.

We investigate generalized Bochner-Riesz means at the critical index on spaces generated by smooth blocks and give some approximation theorems.

In this paper, we analyze multi-dimensional quasi-asymptotically $c$-almost periodic functions and their Stepanov generalizations as well as multi-dimensional Weyl $c$-almost periodic type functions. We also analyze several important subclasses of the class of multi-dimensional quasi-asymptotically $c$-almost periodic functions and reconsider the notion of semi-$c$-periodicity in the multi-dimensional setting, working in the general framework of Lebesgue spaces with variable exponent. We provide certain...

The construction of generalized continuous wavelet transforms on locally compact abelian groups A from quasi-regular representations of a semidirect product group G = A ⋊ H acting on L²(A) requires the existence of a square-integrable function whose Plancherel transform satisfies a Calderón-type resolution of the identity. The question then arises under what conditions such square-integrable functions exist. The existing literature on this subject leaves a gap between sufficient and necessary criteria....